Question

problem. 5 : compute the following derivative:
$\frac{d}{dx}(x^3 - 3x^2 - 24x + 80) = $
problem. 6 : compute the first and second derivatives for the function
$f(x) = -3x^3 - 45x^2 - 225x - 375$.
$f(x) = $
$f(x) = $

Explanation:

Problem 5

Step1: Apply power rule to each term

The power rule is $\frac{d}{dx}(x^n)=nx^{n-1}$, and derivative of constant is 0.
$\frac{d}{dx}(x^3) = 3x^{2}$, $\frac{d}{dx}(-3x^2)=-6x$, $\frac{d}{dx}(-24x)=-24$, $\frac{d}{dx}(80)=0$

Step2: Sum the derivatives

$3x^2 - 6x - 24 + 0$

Problem 6 (First derivative)

Step1: Apply power rule to each term

$\frac{d}{dx}(-3x^3)=-9x^2$, $\frac{d}{dx}(-45x^2)=-90x$, $\frac{d}{dx}(-225x)=-225$, $\frac{d}{dx}(-375)=0$

Step2: Sum the derivatives

$-9x^2 - 90x - 225 + 0$

Problem 6 (Second derivative)

Step1: Apply power rule to $f'(x)$

$\frac{d}{dx}(-9x^2)=-18x$, $\frac{d}{dx}(-90x)=-90$, $\frac{d}{dx}(-225)=0$

Step2: Sum the derivatives

$-18x - 90 + 0$

Answer:

Problem 5: $3x^2 - 6x - 24$
Problem 6: $f'(x)=-9x^2 - 90x - 225$
Problem 6: $f''(x)=-18x - 90$